In this paper, we examine the invariant subspaces (under the operator f -->z f) M of the Bergman space pa (G\T) (where 1 < p < 2, G is a bounded region in C containing D, T is the unit circle, and D is the unit disk) which contain the characteristic functions xD and xG, i.e. the constant functions on the components of G\T. We will show that such M are in one-to-one correspondence with the invariant subspaces of the analytic Besov space ABq (q is the conjugate index to p) and then use results of Shirokov to describe such M. When p ≥ 2 the situation becomes more complicated and capacity considerations are needed.
Copyright © 1994 Indiana University Mathematics Journal. This article first appeared in Indiana University Mathematics Journal 43:4 (1994), 1297-1319.
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Ross, William T. "Analytic Besov Spaces and Invariant Subspaces of Bergman Spaces." Indiana University Mathematics Journal 43, no. 4 (1994): 1297-1319. doi:10.1512/iumj.1994.43.43056.